Topological characterization of neuronal arbor morphology via sequence representation: II--global alignment

Automated Analysis of Neuronal Morphology in 2D Fluorescence Micrographs through an Unsupervised Semantic Segmentation of Neurons

Brain function emerges from a highly complex network of specialized cells that are interlinked by billions of synapses. The synaptic connectivity between neurons is established between the elongated processes of their axons and dendrites or, together, neurites. To establish these connections, cellular neurites have to grow in highly specialized, cell-type dependent patterns covering extensive distances and connecting with thousands of other neurons. The outgrowth and branching of neurites are tightly controlled during development and are a commonly used functional readout of imaging in the neurosciences. Manual analysis of neuronal morphology from microscopy images, however, is very time intensive and prone to bias. Most automated analyses of neurons rely on reconstruction of the neuron as a whole without a semantic analysis of each neurite. A fully-automated classification of all neurites still remains unavailable in open-source software. Here we present a standalone, GUI-based software for batch-quantification of neuronal morphology in two-dimensional fluorescence micrographs of cultured neurons with minimal requirements for user interaction. Single neurons are first reconstructed into binarized images using a Hessian-based segmentation algorithm to detect thin neurite structures combined with intensity- and shape-based reconstruction of the cell body. Neurites are then classified into axon, dendrites and their branches of increasing order using a geodesic distance transform of the cell skeleton. The software was benchmarked against a published dataset and reproduced the phenotype observed after manual annotation. Our tool promises accelerated and improved morphometric studies of neuronal morphology by allowing for consistent and automated analysis of large datasets.

Large scale similarity search across digital reconstructions of neural morphology

Most functions of the nervous system depend on neuronal and glial morphology. Continuous advances in microscopic imaging and tracing software have provided an increasingly abundant availability of 3D reconstructions of arborizing dendrites, axons, and processes, allowing their detailed study. However, efficient, large-scale methods to rank neural morphologies by similarity to an archetype are still lacking. Using the NeuroMorpho.Org database, we present a similarity search software enabling fast morphological comparison of hundreds of thousands of neural reconstructions from any species, brain regions, cell types, and preparation protocols. We compared the performance of different morphological measurements: 1) summary morphometrics calculated by L-Measure, 2) persistence vectors, a vectorized descriptor of branching structure, 3) the combination of the two. In all cases, we also investigated the impact of applying dimensionality reduction using principal component analysis (PCA). We assessed qualitative performance by gauging the ability to rank neurons in order of visual similarity. Moreover, we quantified information content by examining explained variance and benchmarked the ability to identify occasional duplicate reconstructions of the same specimen. We also compared two different methods for selecting the number of principal components using this benchmark. The results indicate that combining summary morphometrics and persistence vectors with applied PCA using maximum likelihood based automatic dimensionality selection provides an information rich characterization that enables efficient and precise comparison of neural morphology. We have deployed the similarity search as open-source online software both through a user-friendly graphical interface and as an API for programmatic access.

Exploring highly reliable substructures in auto-reconstructions of a neuron

The digital reconstruction of a neuron is the most direct and effective way to investigate its morphology. Many automatic neuron tracing methods have been proposed, but without manual check it is difficult to know whether a reconstruction or which substructure in a reconstruction is accurate. For a neuron's reconstructions generated by multiple automatic tracing methods with different principles or models, their common substructures are highly reliable and named individual motifs. In this work, we propose a Vaa3D-based method called Lamotif to explore individual motifs in automatic reconstructions of a neuron. Lamotif utilizes the local alignment algorithm in BlastNeuron to extract local alignment pairs between a specified objective reconstruction and multiple reference reconstructions, and combines these pairs to generate individual motifs on the objective reconstruction. The proposed Lamotif is evaluated on reconstructions of 163 multiple species neurons, which are generated by four state-of-the-art tracing methods. Experimental results show that individual motifs are almost on corresponding gold standard reconstructions and have much higher precision rate than objective reconstructions themselves. Furthermore, an objective reconstruction is mostly quite accurate if its individual motifs have high recall rate. Individual motifs contain common geometry substructures in multiple reconstructions, and can be used to select some accurate substructures from a reconstruction or some accurate reconstructions from automatic reconstruction dataset of different neurons.

A Systematic Evaluation of Interneuron Morphology Representations for Cell Type Discrimination

Quantitative analysis of neuronal morphologies usually begins with choosing a particular feature representation in order to make individual morphologies amenable to standard statistics tools and machine learning algorithms. Many different feature representations have been suggested in the literature, ranging from density maps to intersection profiles, but they have never been compared side by side. Here we performed a systematic comparison of various representations, measuring how well they were able to capture the difference between known morphological cell types. For our benchmarking effort, we used several curated data sets consisting of mouse retinal bipolar cells and cortical inhibitory neurons. We found that the best performing feature representations were two-dimensional density maps, two-dimensional persistence images and morphometric statistics, which continued to perform well even when neurons were only partially traced. Combining these feature representations together led to further performance increases suggesting that they captured non-redundant information. The same representations performed well in an unsupervised setting, implying that they can be suitable for dimensionality reduction or clustering.

Density Visualization Pipeline: A Tool for Cellular and Network Density Visualization and Analysis

Neuron classification is an important component in analyzing network structure and quantifying the effect of neuron topology on signal processing. Current quantification and classification approaches rely on morphology projection onto lower-dimensional spaces. In this paper a 3D visualization and quantification tool is presented. The Density Visualization Pipeline (DVP) computes, visualizes and quantifies the density distribution, i.e., the "mass" of interneurons. We use the DVP to characterize and classify a set of GABAergic interneurons. Classification of GABAergic interneurons is of crucial importance to understand on the one hand their various functions and on the other hand their ubiquitous appearance in the neocortex. 3D density map visualization and projection to the one-dimensional x, y, z subspaces show a clear distinction between the studied cells, based on these metrics. The DVP can be coupled to computational studies of the behavior of neurons and networks, in which network topology information is derived from DVP information. The DVP reads common neuromorphological file formats, e.g., Neurolucida XML files, NeuroMorpho.org SWC files and plain ASCII files. Full 3D visualization and projections of the density to 1D and 2D manifolds are supported by the DVP. All routines are embedded within the visual programming IDE VRL-Studio for Java which allows the definition and rapid modification of analysis workflows.

Objective Morphological Classification of Neocortical Pyramidal Cells

A consensus on the number of morphologically different types of pyramidal cells (PCs) in the neocortex has not yet been reached, despite over a century of anatomical studies, due to the lack of agreement on the subjective classifications of neuron types, which is based on expert analyses of neuronal morphologies. Even for neurons that are visually distinguishable, there is no common ground to consistently define morphological types. The objective classification of PCs can be achieved with methods from algebraic topology, and the dendritic arborization is sufficient for the reliable identification of distinct types of cortical PCs. Therefore, we objectively identify 17 types of PCs in the rat somatosensory cortex. In addition, we provide a solution to the challenging problem of whether 2 similar neurons belong to different types or to a continuum of the same type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or continuous morphological variations of each other.

Edit Distance between Merge Trees

Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of sub-level and super-level sets in a scalar field. Estimating the similarity between merge trees is an important problem with applications to feature-directed visualization of time-varying data. We present an approach based on tree edit distance to compare merge trees. The comparison measure satisfies metric properties, it can be computed efficiently, and the cost model for the edit operations is both intuitive and captures well-known properties of merge trees. Experimental results on time-varying scalar fields, 3D cryo electron microscopy data, shape data, and various synthetic datasets show the utility of the edit distance towards a feature-driven analysis of scalar fields.

NeuroPath2Path: Classification and elastic morphing between neuronal arbors using path-wise similarity

Neuron shape and connectivity affect function. Modern imaging methods have proven successful at extracting morphological information. One potential path to achieve analysis of this morphology is through graph theory. Encoding by graphs enables the use of high throughput informatic methods to extract and infer brain function. However, the application of graph-theoretic methods to neuronal morphology comes with certain challenges in term of complex subgraph matching and the difficulty in computing intermediate shapes in between two imaged temporal samples. Here we report a novel, efficacious graph-theoretic method that rises to the challenges. The morphology of a neuron, which consists of its overall size, global shape, local branch patterns, and cell-specific biophysical properties, can vary significantly with the cell's identity, location, as well as developmental and physiological state. Various algorithms have been developed to customize shape based statistical and graph related features for quantitative analysis of neuromorphology, followed by the classification of neuron cell types using the features. Unlike the classical feature extraction based methods from imaged or 3D reconstructed neurons, we propose a model based on the rooted path decomposition from the soma to the dendrites of a neuron and extract morphological features from each constituent path. We hypothesize that measuring the distance between two neurons can be realized by minimizing the cost of continuously morphing the set of all rooted paths of one neuron to another. To validate this claim, we first establish the correspondence of paths between two neurons using a modified Munkres algorithm. Next, an elastic deformation framework that employs the square root velocity function is established to perform the continuous morphing, which, as an added benefit, provides an effective visualization tool. We experimentally show the efficacy of NeuroPath2Path, NeuroP2P, over the state of the art.

A Topological Representation of Branching Neuronal Morphologies

Many biological systems consist of branching structures that exhibit a wide variety of shapes. Our understanding of their systematic roles is hampered from the start by the lack of a fundamental means of standardizing the description of complex branching patterns, such as those of neuronal trees. To solve this problem, we have invented the Topological Morphology Descriptor (TMD), a method for encoding the spatial structure of any tree as a "barcode", a unique topological signature. As opposed to traditional morphometrics, the TMD couples the topology of the branches with their spatial extents by tracking their topological evolution in 3-dimensional space. We prove that neuronal trees, as well as stochastically generated trees, can be accurately categorized based on their TMD profiles. The TMD retains sufficient global and local information to create an unbiased benchmark test for their categorization and is able to quantify and characterize the structural differences between distinct morphological groups. The use of this mathematically rigorous method will advance our understanding of the anatomy and diversity of branching morphologies.

Modelling brain-wide neuronal morphology via rooted Cayley trees

Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent features of brain-wide neuronal morphology. We observe that axonal trees are self-affine while dendritic trees are self-similar. We also show that tree size appear to be random, independent of the number of dendrites within single neurons. Moreover, we consider inhomogeneous branching model which stochastically generates rooted 3-Cayley trees for the brain-wide neuron topology. Based on estimated order-dependent branching probability from actual axonal and dendritic trees, our inhomogeneous model quantitatively captures a number of topological features including size and shape of both axons and dendrites. This sheds lights on a universal mechanism behind the topological formation of brain-wide axonal and dendritic trees.

Metrics for comparing neuronal tree shapes based on persistent homology

As more and more neuroanatomical data are made available through efforts such as NeuroMorpho.Org and FlyCircuit.org, the need to develop computational tools to facilitate automatic knowledge discovery from such large datasets becomes more urgent. One fundamental question is how best to compare neuron structures, for instance to organize and classify large collection of neurons. We aim to develop a flexible yet powerful framework to support comparison and classification of large collection of neuron structures efficiently. Specifically we propose to use a topological persistence-based feature vectorization framework. Existing methods to vectorize a neuron (i.e, convert a neuron to a feature vector so as to support efficient comparison and/or searching) typically rely on statistics or summaries of morphometric information, such as the average or maximum local torque angle or partition asymmetry. These simple summaries have limited power in encoding global tree structures. Based on the concept of topological persistence recently developed in the field of computational topology, we vectorize each neuron structure into a simple yet informative summary. In particular, each type of information of interest can be represented as a descriptor function defined on the neuron tree, which is then mapped to a simple persistence-signature. Our framework can encode both local and global tree structure, as well as other information of interest (electrophysiological or dynamical measures), by considering multiple descriptor functions on the neuron. The resulting persistence-based signature is potentially more informative than simple statistical summaries (such as average/mean/max) of morphometric quantities-Indeed, we show that using a certain descriptor function will give a persistence-based signature containing strictly more information than the classical Sholl analysis. At the same time, our framework retains the efficiency associated with treating neurons as points in a simple Euclidean feature space, which would be important for constructing efficient searching or indexing structures over them. We present preliminary experimental results to demonstrate the effectiveness of our persistence-based neuronal feature vectorization framework.

BlastNeuron for Automated Comparison, Retrieval and Clustering of 3D Neuron Morphologies

Characterizing the identity and types of neurons in the brain, as well as their associated function, requires a means of quantifying and comparing 3D neuron morphology. Presently, neuron comparison methods are based on statistics from neuronal morphology such as size and number of branches, which are not fully suitable for detecting local similarities and differences in the detailed structure. We developed BlastNeuron to compare neurons in terms of their global appearance, detailed arborization patterns, and topological similarity. BlastNeuron first compares and clusters 3D neuron reconstructions based on global morphology features and moment invariants, independent of their orientations, sizes, level of reconstruction and other variations. Subsequently, BlastNeuron performs local alignment between any pair of retrieved neurons via a tree-topology driven dynamic programming method. A 3D correspondence map can thus be generated at the resolution of single reconstruction nodes. We applied BlastNeuron to three datasets: (1) 10,000+ neuron reconstructions from a public morphology database, (2) 681 newly and manually reconstructed neurons, and (3) neurons reconstructions produced using several independent reconstruction methods. Our approach was able to accurately and efficiently retrieve morphologically and functionally similar neuron structures from large morphology database, identify the local common structures, and find clusters of neurons that share similarities in both morphology and molecular profiles.

Topological characterization of neuronal arbor morphology via sequence representation: I--motif analysis

BACKGROUND

The morphology of neurons offers many insights into developmental processes and signal processing. Numerous reports have focused on metrics at the level of individual branches or whole arbors; however, no studies have attempted to quantify repeated morphological patterns within neuronal trees. We introduce a novel sequential encoding of neurite branching suitable to explore topological patterns.

RESULTS

Using all possible branching topologies for comparison we show that the relative abundance of short patterns of up to three bifurcations, together with overall tree size, effectively capture the local branching patterns of neurons. Dendrites and axons display broadly similar topological motifs (over-represented patterns) and anti-motifs (under-represented patterns), differing most in their proportions of bifurcations with one terminal branch and in select sub-sequences of three bifurcations. In addition, pyramidal apical dendrites reveal a distinct motif profile.

CONCLUSIONS

The quantitative characterization of topological motifs in neuronal arbors provides a thorough description of local features and detailed boundaries for growth mechanisms and hypothesized computational functions.